Le groupe d'automorphismes du groupe modulaire

Tchangang, Tambekou Roger

Annales de l'Institut Fourier, Tome 37 (1987) no. 2, pp. 19-31.

Le but de cet article est de donner une autre démonstration plus simple du théorème d’Ivanov (Théorème 1) qui assure que le groupe $M_g^{*}$ de toutes les difféotopies d’une surface $F_g$ orientable et fermée de genre $g\ge 2$ est complet. En étudiant l’action d’un automorphisme quelconque du groupe $M_g^{*}$ sur les difféotopies d’ordre fini, on montre que les involutions hyperelliptiques sont globalement préservées. Le théorème d’Ivanov est alors une conséquence d’un résultat de Dyer et Grossmann qui affirm que le groupe des automorphismes extérieurs du groupe des tresses est isomorphe à ${\mathbf{Z}}_2$. La démonstration s’adapte aussi au cas des surfaces pointées.

We give a new and simple proof of the fact that the full mapping class group $M_g^{*}$ of a closed oriented surface of genus $g\ge 3$ is complete (this is known as Ivanov’s theorem). In studying the action of an automorphism on finite subgroups of $M_g^{*}$, one remarks that hyperelliptic involutions are mapped onto themselves. The argument also uses Dyer and Grossmann’s result asserting that the outer group of automorphism of the braid group ${\mathbf{B}}_n$ is isomorphic to ${\mathbf{Z}}_2$. The proof extends to the case of surfaces with one puncture. Unfortunately, our method doesn’t prove the theorem of Ivanov in the case of surfaces with finitely many punctures.

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